Abstract
We study the asymptotics of the global fluctuations for the difference between two adjacent levels in the \(\beta \)–Jacobi corners process (multilevel and general \(\beta \) extension of the classical Jacobi ensemble of random matrices). The limit is identified with the derivative of the 2d Gaussian free field. Our main tools are integral forms for the (Macdonald-type) difference operators originating from the shuffle algebra.
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Acknowledgements
The authors would like to thank Alexei Borodin and Alexey Bufetov for helpful discussions. We are very grateful to the two anonymous referees, whose feedback led to many important improvements in the text. VG was partially supported by the NSF Grants DMS-1407562, DMS-1664619, and by the Sloan Research Fellowship. LZ was partially supported by Undergraduate Research Opportunity Program (UROP) in Massachusetts Institute of Technology.
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Appendices
Appendices
Dimension reduction identities
In this appendix we discuss integral identities, which are widely used in proofs in the main text. A special case (\(m=1\)) of the following result was communicated to the authors by Alexei Borodin, and we present our own proof here.
For any positive integer n, let \(\sigma _n\) denote the cycle \((12\cdots n)\), and let \(S^{cyc}(n)\) denote the n-element subgroup of the symmetric group spanned by \(\sigma _n\).
Theorem A.1
Let \(n\ge 2\), and \(f_1, \ldots , f_n : \mathbb {C} \rightarrow \mathbb {C}\) be meromorphic with possible poles at \(\{\mathfrak {p}_1, \ldots , \mathfrak {p}_m \}\). Then we have the identity
where the contours in both sides are positively oriented, enclosing \(\{\mathfrak {p}_1, \ldots , \mathfrak {p}_m \}\), and for the left hand side we require \(|u_1| \ll \cdots \ll |u_n|\).
Proof
Let \(\mathfrak {C}_1, \ldots , \mathfrak {C}_{2n-1}\) be closed paths around \(\{\mathfrak {p}_1, \ldots , \mathfrak {p}_m \}\), and each \(\mathfrak {C}_i\) is inside \(\mathfrak {C}_{i+1}\), \(1\le i \le 2n-2\). Also, to simplify notations, set \(f_{n+t} = f_t\) and \(u_{n+t} = u_t\) for any \(1 \le t \le n-1\). Then the left hand side of (A.1) can be written as
When \(n=2\), we have
where \(\oint _{\mathfrak {C}_{3} - \mathfrak {C}_{1}}\) is a notation for the difference of integrals over \(\mathfrak {C}_3\) and \(\mathfrak {C}_1\). Further, (A.3) equals to
since as a function of \(u_1\), \(\frac{f_1(u_1) f_2(u_2)}{u_1 - u_2} \) has a single pole at \(u_2\) between \(\mathfrak {C}_{3}\) and \(\mathfrak {C}_{1}\); and the residue at this pole equals \(f_1(u_2)f_2(u_2)\). This proves the case of \(n=2\).
When \(n\ge 3\), we argue by induction and assume that Theorem A.1 is true for \(n-1\). For any \(1 \le t \le n-1\), we have that
Now we can move the contours of \(u_1, \ldots , u_t\) from \(\mathfrak {C}_{n+1}, \ldots , \mathfrak {C}_{n+t}\) to \(\mathfrak {C}_{1}, \ldots , \mathfrak {C}_{t}\), respectively. We move the contours one by one starting from \(u_1\), and each move is across \(\mathfrak {C}_{t+1}, \ldots , \mathfrak {C}_{n}\). For \(u_1 (= u_{n+1})\), the only pole between \(\mathfrak {C}_{n+1}\) and \(\mathfrak {C}_{1}\) is \(u_n\); for any \(u_i\), \(1 < i \le t\), there is no pole between \(\mathfrak {C}_{n+i}\) and \(\mathfrak {C}_{i}\). Thus we have that
Notice that (taking into account that \(u_{n+t} = u_t\))
and by the induction assumption (applied to \(f_2, \ldots , f_{n-1}, f_n f_1\)), we have that
\(\square \)
Corollary A.2
Let s be a positive integer. Let f, \(g_1, \ldots , g_s\) be meromorphic functions with possible poles at \(\{\mathfrak {p}_1, \ldots , \mathfrak {p}_m \}\). Then for \(n\ge 2\),
where the contours in both sides are around all of \(\{\mathfrak {p}_1, \ldots , \mathfrak {p}_m \}\), and for the left hand side we require \(|u_1| \ll \cdots \ll |u_n|\).
Proof
Take disjoint sets \(U_1, \ldots , U_n\), with \(\bigcup _{i=1}^n U_i = \{1, \ldots , s\}\) (some of which might be empty). In Theorem A.1 we let \(f_{i} = f\prod _{j\in U_{i} } g_j\) for each \(1 \le i \le n\), and get
Summing over all \(n^s\) partitions \(U_1, \ldots , U_n\) of \(\{1, \ldots , s\}\) into n disjoint sets, we obtain (A.9). \(\square \)
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Gorin, V., Zhang, L. Interlacing adjacent levels of \(\beta \)–Jacobi corners processes. Probab. Theory Relat. Fields 172, 915–981 (2018). https://doi.org/10.1007/s00440-017-0823-8
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DOI: https://doi.org/10.1007/s00440-017-0823-8
Mathematics Subject Classification
- 33C45
- 60B20
- 60G60